Question

What is the standard form of `y= (4x-2x^2)(-4x+3) -(x-4)^3`?

Answer 1

`7x^3-10x^2-36x+64`

Explanation:

In order to complete this problem (and make it easier on ourselves), we should split it into two parts. Let's start with the first part, `(4x-2x^2)(-4x+3)`. Using FOIL (First, Outsides, Insides, Last), we can simplify this down:

`(4x)(-4x)+(4x)(3)+(-2x^2)(-4x)+(-2x^2)(3)`
`-16x^2+12x+8x^3-6x^2`

We don't need to write this in standard form quite yet as we still have the other half of the equation, `(x-4)^3`. The formula for the multiplication of `(a-b)^3` is `a^3+3a^2b+3ab^2+b^3`. Let's use this as a guideline for our multiplication.

`(x-4)^3`
`x^3+(3)(x^2)(-4)+(3)(x)(-4)^2+(-4)^3`
`x^3-12x^2+48x-64`

Let's now combine our two parts to create one answer:

`-16x^2+12x+8x^3-6x^2-(x^3-12x^2+48x-64)`
`-16x^2+12x+8x^3-6x^2-x^3+12x^2-48x+64`
`8x^3-x^3-16x^2-6x^2+12x^2+12x-48x+64`
`7x^3-10x^2-36x+64`